Mathematics is effective in science. Wigner (1960: 14) regards this effectiveness as magical: “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” The prudent reply that it is surely not very scientific to base scientific reasoning on miracles. A more rational alternative says that mathematics is effective in science because physical reality is grounded in mathematical reality.

The Effectiveness Argument goes like this: (1) Mathematics is effective in science. (2) The best explanation for this effectiveness is that physical reality is grounded in mathematical reality. (3) So, by inference to the best explanation, all physical reality, including our universe, is grounded in mathematical reality – in pure mathematics.

The second premise in the Effectiveness Argument is supported by a variety of writers. Dipert (1997: 332) argues that “the very possibility of a clear understanding of the world requires the possibility that it is a simple mathematical structure”. Steiner (1998: 4 – 5) puts it even more powerfully like this:

The strategy physicists pursued . . . to guess at the laws of nature, was a Pythagorean strategy: they used the relations between the structures and even the notations of mathematics to frame analogies and guess according to those analogies. The strategy succeeded. . . . The success of the Pythagorean strategy might lead the reader to conceptual Pythagoreanism, the view that the ultimate properties or ‘real essences’ of things are none other than the mathematical structures and their relations. More radically, one might adopt metaphysical Pythagoreanism, which simply identifies the Universe or the things in it with mathematical objects or structures. (Some physicists write as though an elementary particle just ‘is’ an irreducible group representation, or even that the entire universe is.)

Steiner (1998: ch. 4) brilliantly discusses many examples in which the pythagorean strategy of identifying physical things with mathematical things is successful. His cases include: Maxwell’s study of electromagnetism; Schroedinger’s study of wave mechanics; Dirac’s study of the positron; Schwarzschild’s solution for the equations of general relativity (i.e. black holes); Heisenberg’s study of the symmetries of nucleons; Kemmer’s study of pions; Gell-Mann’s and Ne’eman’s study of particle systems with unitary spin and the consequent discovery of quarks; Einstein’s inference of the field equations for general relativity; the Heisenberg-Born-Jordan derivation of matrix mechanics; Schroedinger’s derivation of the Klein-Gordon equation; the derivation of the Yang-Mills equation; the study of analytic continuations in crossing symmetries.

As a continuation of Steiner’s reasoning, Tegmark (1998: 44) says: “the usefulness of mathematics for describing the physical world is a natural consequence of the fact that the latter is a mathematical structure.” Accordingly, Tegmark (1998: 46-47) simply collapses the distinction between mathematical and physical existence:

One might say that wherever there is light, there are associated ripples in the electromagnetic field. But the modern view is that light is the ripples. One might say that wherever there is matter, there are associated ripples in the metric known as curvature. But Eddington’s view is that matter is the ripples. One might say that wherever there is physical existence, there is an associated mathematical structure. But according to our TOE [theory of everything], physical existence is mathematical existence. (The italics are Tegmark’s.)

Dipert, R. (1997) The mathematical structure of the world: The world as graph. Journal of Philosophy 94 (7), 329-358.

Steiner, M. (1998) The Applicability of Mathematics as a Philosophical Problem. Cambridge, MA: Harvard University Press.